Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain why \((5,-2)\) and \((-2,5)\) do not represent the same point.

Step-by-Step Solution

Answer

The points \((5,-2)\) and \((-2,5)\) represent different locations on the Cartesian plane because their x-coordinates \(5 \neq -2\) and their y-coordinates \(-2 \neq 5\). Therefore, the points \((5,-2)\) and \((-2,5)\) do not represent the same point.

Step 1: Understand the coordinate system

A point in the two-dimensional Cartesian coordinate system is represented by an ordered pair of numbers \((x, y)\), called coordinates. This pair represents the horizontal (x) and vertical (y) distances from the origin, usually denoted as \((0,0)\)

Step 2: Identify the coordinates of the two points

The two given points are \((5,-2)\) and \((-2,5)\). Here, for the point \((5,-2)\), the x-coordinate is 5 and the y-coordinate is -2 whereas for the point \((-2,5)\), the x-coordinate is -2 and the y-coordinate is 5.

Step 3: Determine if the two points are the same

Two points are the same if and only if both of their x-coordinates are the same and their y-coordinates are the same. Comparing the two points, the first point has an x-coordinate of 5, which doesn't match the x-coordinate of the second point, -2. Similarly, the first point has a y-coordinate of -2, while the second point has a y-coordinate of 5. Since neither the x-coordinates nor the y-coordinates are the same, the points \((5,-2)\) and \((-2,5)\) are not the same.